The Stacks project

Lemma 10.127.8. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } R_\lambda $ is a directed colimit of rings. Then the category of finitely presented $R$-algebras is the colimit of the categories of finitely presented $R_\lambda $-algebras. More precisely

  1. Given a finitely presented $R$-algebra $A$ there exists a $\lambda \in \Lambda $ and a finitely presented $R_\lambda $-algebra $A_\lambda $ such that $A \cong A_\lambda \otimes _{R_\lambda } R$.

  2. Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-algebras $A_\lambda , B_\lambda $, and an $R$-algebra map $\varphi : A_\lambda \otimes _{R_\lambda } R \to B_\lambda \otimes _{R_\lambda } R$, then there exists a $\mu \geq \lambda $ and an $R_\mu $-algebra map $\varphi _\mu : A_\lambda \otimes _{R_\lambda } R_\mu \to B_\lambda \otimes _{R_\lambda } R_\mu $ such that $\varphi = \varphi _\mu \otimes 1_ R$.

  3. Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-algebras $A_\lambda , B_\lambda $, and $R_\lambda $-algebra maps $\varphi _\lambda , \psi _\lambda : A_\lambda \to B_\lambda $ such that $\varphi \otimes 1_ R = \psi \otimes 1_ R$, then $\varphi \otimes 1_{R_\mu } = \psi \otimes 1_{R_\mu }$ for some $\mu \geq \lambda $.

Proof. To prove (1) choose a presentation $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. We can choose a $\lambda \in \Lambda $ and elements $f_{\lambda , j} \in R_\lambda [x_1, \ldots , x_ n]$ mapping to $f_ j \in R[x_1, \ldots , x_ n]$. Then we simply let $A_\lambda = R_\lambda [x_1, \ldots , x_ n]/(f_{\lambda , 1}, \ldots , f_{\lambda , m})$.

Parts (2) and (3) follow from Lemma 10.127.7. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05N9. Beware of the difference between the letter 'O' and the digit '0'.